Mother's Day, Red Sox, and Jordan's Furniture

In honor of Mother’s Day, I’d like to do a problem suggested by my mom. She’s a huge Red Sox fan, and she’s curious about the Jordan’s Furniture promotion going on at Fenway Park. According to Boston.com,

“customers who buy anything at one of the regional chain's stores from April 1 through May 2 won't have to pay for their purchase if a Red Sox player hits the image of a Jordan's Furniture baseball on the company sign at Fenway Park between July 15 and the end of the season Oct. 3”

Mom would like to know, “What are the odds that a Red Sox player will hit the Jordan’s sign?"

There are 35 Red Sox home games between July 15 and the end of the season. As of today, the Sox have hit 42 homeruns through 31 games, which suggests they’d hit about 47 through 35 games.

If we assume homeruns have an equal probability of being hit to left, center, and right field, then we can calculate the fraction of homeruns hit towards the Jordan’s sign. Since the foul lines are at right angles to each other, homeruns can only be hit in one quadrant of a circle. Any other ball that left the park would be foul. A quadrant of a circle has an angle of Pi/2~1.57 radians or 90°. The Jordan’s sign is 12 ft long and at 421 ft from home plate, so it spans and angle of about 0.0284 radians. From this we can compute the probability that a homerun is hit at the correct angle,

probability of correct angle = (angular width of sign) / (total angular width)

= (0.0284 radians) / (1.57 radians)

= 0.018.

This means a homerun has about a 1.8% chance of being hit in the direction of the Jordan’s sign. However, this doesn’t mean it will be hit at the correct height. It’s difficult to say exactly what fraction will be hit at the right height, but as a rough estimate, I’ll assume one of every five homeruns will be at the right height to hit the sign. This means 0.4% of homeruns will hit the sign, or, equivalently, 99.6% of homeruns will not hit the sign. If there are 47 homeruns, the probability that none of them will hit the sign is

(0.996)⁴⁷ = 0.83,

or, about 83%. This means there’s about a 17% chance that one of the homeruns will hit the sign.

Happy Mother’s Day, Mom!